GATE CS 2026 Set 2 Q.31: Linear Algebra

Consider the system of linear equations given below.

ax + y = b

16x + ay = 24

Suppose the values of $a$ and $b$ are chosen such that the system of linear equations produce multiple solutions. Then the product of $a$ and $b$ is __________. (answer in integer)

Your answer

Official answer and solution

Official answer: 24
From: Official GATE 2026 answer key

Solution

Use the condition $\frac{a}{16} = \frac{1}{a} = \frac{b}{24}$ for two linear equations in two variables to have infinitely many solutions.

For a $2 \times 2$ linear system to have multiple (infinitely many) solutions, the two equations must be proportional:

\frac{a}{16} = \frac{1}{a} = \frac{b}{24}

From $\frac{1}{a} = \frac{b}{24}$ , cross-multiplying directly gives:

a \cdot b = 24

(Specifically, $a^2 = 16 \Rightarrow a = \pm 4$ and $b = \pm 6$ , and in both cases $ab = 24$ .)

Final answer

Common trap

Solving only for $a = 4$ and forgetting that the question asks for the product $ab$ , or overlooking that $ab = 24$ holds for both $a = 4$ and $a = -4$ .

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2026 Computer Science and Information Technology - Session 2 (CS2)
Question: 31 of 65
Section: Computer Science and Information Technology

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